Joint Entrance Examination

Graduate Aptitude Test in Engineering

Geomatics Engineering Or Surveying

Engineering Mechanics

Hydrology

Transportation Engineering

Strength of Materials Or Solid Mechanics

Reinforced Cement Concrete

Steel Structures

Irrigation

Environmental Engineering

Engineering Mathematics

Structural Analysis

Geotechnical Engineering

Fluid Mechanics and Hydraulic Machines

General Aptitude

1

If one real root of the quadratic equation 81x^{2} + kx + 256 = 0 is cube of the other root, then a value of k is

A

$$-$$ 81

B

$$-$$ 300

C

100

D

144

81x^{2} + kx + 256 = 0 ; x = $$\alpha $$, $$\alpha $$^{3}

$$ \Rightarrow $$ $$\alpha $$^{4} = $${{256} \over {81}}$$ $$ \Rightarrow $$ $$\alpha $$ = $$ \pm $$ $${{4} \over {3}}$$

Now $$-$$ $${k \over {81}}$$ = $$\alpha $$ + $$\alpha $$^{3} = $$ \pm $$ $${{100} \over {27}}$$

$$ \Rightarrow $$ k = $$ \pm $$300

$$ \Rightarrow $$ $$\alpha $$

Now $$-$$ $${k \over {81}}$$ = $$\alpha $$ + $$\alpha $$

$$ \Rightarrow $$ k = $$ \pm $$300

2

Let $$\alpha $$ and $$\beta $$ be the roots of the quadratic equation x^{2}
sin $$\theta $$ – x(sin $$\theta $$ cos $$\theta $$ + 1) + cos $$\theta $$ = 0 (0 < $$\theta $$ < 45^{o}), and $$\alpha $$ < $$\beta $$. Then $$\sum\limits_{n = 0}^\infty {\left( {{\alpha ^n} + {{{{\left( { - 1} \right)}^n}} \over {{\beta ^n}}}} \right)} $$ is equal to :

A

$${1 \over {1 + \cos \theta }} + {1 \over {1 - \sin \theta }}$$

B

$${1 \over {1 - \cos \theta }} + {1 \over {1 + \sin \theta }}$$

C

$${1 \over {1 - \cos \theta }} - {1 \over {1 + \sin \theta }}$$

D

$${1 \over {1 + \cos \theta }} - {1 \over {1 - \sin \theta }}$$

D = (1 + sin$$\theta $$ cos$$\theta $$)^{2} $$-$$ 4sin$$\theta $$cos$$\theta $$ = (1 $$-$$ sin$$\theta $$ cos$$\theta $$)^{2}

$$ \Rightarrow $$ roots are $$\beta $$ = cosec$$\theta $$ and $$\alpha $$ = cos$$\theta $$

$$\sum\limits_{n = 0}^\infty {\left( {{\alpha ^n} + {{\left( { - {1 \over \beta }} \right)}^n}} \right)} = \sum\limits_{n = 0}^\infty {{{\left( {\cos \theta } \right)}^n}} + \sum\limits_{n = 0}^n {{{\left( { - \sin \theta } \right)}^n}} $$

$$ = {1 \over {1 - \cos \theta }} + {1 \over {1 + \sin \theta }}$$

$$ \Rightarrow $$ roots are $$\beta $$ = cosec$$\theta $$ and $$\alpha $$ = cos$$\theta $$

$$\sum\limits_{n = 0}^\infty {\left( {{\alpha ^n} + {{\left( { - {1 \over \beta }} \right)}^n}} \right)} = \sum\limits_{n = 0}^\infty {{{\left( {\cos \theta } \right)}^n}} + \sum\limits_{n = 0}^n {{{\left( { - \sin \theta } \right)}^n}} $$

$$ = {1 \over {1 - \cos \theta }} + {1 \over {1 + \sin \theta }}$$

3

If $$\lambda $$ be the ratio of the roots of the quadratic equation in x, 3m^{2}x^{2} + m(m – 4)x + 2 = 0, then the least value of m for which $$\lambda + {1 \over \lambda } = 1,$$ is

A

$$ - 2 + \sqrt 2 $$

B

4$$-$$3$$\sqrt 2 $$

C

2 $$-$$ $$\sqrt 3 $$

D

4 $$-$$ 2$$\sqrt 3 $$

3m^{2}x^{2} + m(m $$-$$ 4) x + 2 = 0

$$\lambda + {1 \over \lambda } = 1,{\alpha \over \beta } + {\beta \over \alpha } = 1,{\alpha ^2} + {\beta ^2} = \alpha \beta $$

($$\alpha $$ + $$\beta $$)^{2} = 3$$\alpha $$$$\beta $$

$${\left( { - {{m\left( {m - 4} \right)} \over {3{m^2}}}} \right)^2} = {{3\left( 2 \right)} \over {3{m^2}}},{{{{\left( {m - 4} \right)}^2}} \over {9{m^2}}} = {6 \over {3m}}$$

$${\left( {m - 4} \right)^2} = 18,m = 4 \pm \sqrt {18,} \,\,4 \pm 3\sqrt 2 $$

$$\lambda + {1 \over \lambda } = 1,{\alpha \over \beta } + {\beta \over \alpha } = 1,{\alpha ^2} + {\beta ^2} = \alpha \beta $$

($$\alpha $$ + $$\beta $$)

$${\left( { - {{m\left( {m - 4} \right)} \over {3{m^2}}}} \right)^2} = {{3\left( 2 \right)} \over {3{m^2}}},{{{{\left( {m - 4} \right)}^2}} \over {9{m^2}}} = {6 \over {3m}}$$

$${\left( {m - 4} \right)^2} = 18,m = 4 \pm \sqrt {18,} \,\,4 \pm 3\sqrt 2 $$

4

The number of integral values of m for which the quadratic expression, (1 + 2m)x^{2} – 2(1 + 3m)x + 4(1 + m), x $$ \in $$ R, is always positive, is :

A

7

B

8

C

3

D

6

Expression is always positive it

2m + 1 > 0 $$ \Rightarrow $$ m > $$-$$ $${1 \over 2}$$ &

D < 0 $$ \Rightarrow $$ m^{2} $$-$$ 6m $$-$$ 3 < 0

3 $$-$$ $$\sqrt {12} $$ < m < 3 + $$\sqrt {12} $$ . . . . (iii)

$$ \therefore $$ Common interval is

3 $$-$$ $$\sqrt {12} $$ < m < 3 + $$\sqrt {12} $$

$$ \therefore $$ Intgral value of m {0, 1, 2, 3, 4, 5, 6}

2m + 1 > 0 $$ \Rightarrow $$ m > $$-$$ $${1 \over 2}$$ &

D < 0 $$ \Rightarrow $$ m

3 $$-$$ $$\sqrt {12} $$ < m < 3 + $$\sqrt {12} $$ . . . . (iii)

$$ \therefore $$ Common interval is

3 $$-$$ $$\sqrt {12} $$ < m < 3 + $$\sqrt {12} $$

$$ \therefore $$ Intgral value of m {0, 1, 2, 3, 4, 5, 6}

Number in Brackets after Paper Name Indicates No of Questions

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Trigonometric Functions & Equations *keyboard_arrow_right*

Properties of Triangle *keyboard_arrow_right*

Inverse Trigonometric Functions *keyboard_arrow_right*

Complex Numbers *keyboard_arrow_right*

Quadratic Equation and Inequalities *keyboard_arrow_right*

Permutations and Combinations *keyboard_arrow_right*

Mathematical Induction and Binomial Theorem *keyboard_arrow_right*

Sequences and Series *keyboard_arrow_right*

Matrices and Determinants *keyboard_arrow_right*

Vector Algebra and 3D Geometry *keyboard_arrow_right*

Probability *keyboard_arrow_right*

Statistics *keyboard_arrow_right*

Mathematical Reasoning *keyboard_arrow_right*

Functions *keyboard_arrow_right*

Limits, Continuity and Differentiability *keyboard_arrow_right*

Differentiation *keyboard_arrow_right*

Application of Derivatives *keyboard_arrow_right*

Indefinite Integrals *keyboard_arrow_right*

Definite Integrals and Applications of Integrals *keyboard_arrow_right*

Differential Equations *keyboard_arrow_right*

Straight Lines and Pair of Straight Lines *keyboard_arrow_right*

Circle *keyboard_arrow_right*

Conic Sections *keyboard_arrow_right*